fact stringlengths 6 3.84k | type stringclasses 11
values | library stringclasses 32
values | imports listlengths 1 14 | filename stringlengths 20 95 | symbolic_name stringlengths 1 90 | docstring stringlengths 7 20k ⌀ |
|---|---|---|---|---|---|---|
algebraicIndependent_of_finite' (s : Set A)
(hinj : Injective (algebraMap R A))
(H : ∀ t ⊆ s, t.Finite → AlgebraicIndependent R ((↑) : t → A) →
∀ a ∈ s, a ∉ t → Transcendental (adjoin R t) a) :
AlgebraicIndependent R ((↑) : s → A) :=
algebraicIndependent_of_finite_type' hinj fun t hfin h i hi ↦ H _
(by rintro _ ⟨x, _, rfl⟩; exact x.2) (hfin.image _) h.image _ i.2
(mt Subtype.val_injective.mem_set_image.mp hi) | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | algebraicIndependent_of_finite' | Variant of `algebraicIndependent_of_finite` using `Transcendental`. |
sumElim_iff {ι'} {y : ι' → A} : AlgebraicIndependent R (Sum.elim y x) ↔
AlgebraicIndependent R x ∧ AlgebraicIndependent (adjoin R (range x)) y := by
by_cases hx : AlgebraicIndependent R x; swap
· exact ⟨fun h ↦ (hx <| by apply h.comp _ Sum.inr_injective).elim, fun h ↦ (hx h.1).elim⟩
let e := (sumAlgEquiv R ι' ι).trans (mapAlgEquiv _ hx.aevalEquiv)
have : aeval (Sum.elim y x) = ((aeval y).restrictScalars R).comp e.toAlgHom := by
ext (_ | _) <;> simp [e, algebraMap_aevalEquiv]
simp_rw [hx, AlgebraicIndependent, this]; simp | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | sumElim_iff | null |
iff_adjoin_image (s : Set ι) :
AlgebraicIndependent R x ↔ AlgebraicIndependent R (fun i : s ↦ x i) ∧
AlgebraicIndepOn (adjoin R (x '' s)) x sᶜ := by
rw [show x '' s = range fun i : s ↦ x i by ext; simp]
convert ← sumElim_iff
classical apply algebraicIndependent_equiv' ((Equiv.sumComm ..).trans (Equiv.Set.sumCompl ..))
ext (_ | _) <;> rfl | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | iff_adjoin_image | null |
iff_adjoin_image_compl (s : Set ι) :
AlgebraicIndependent R x ↔ AlgebraicIndependent R (fun i : ↥sᶜ ↦ x i) ∧
AlgebraicIndepOn (adjoin R (x '' sᶜ)) x s := by
convert ← iff_adjoin_image _; apply compl_compl | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | iff_adjoin_image_compl | null |
iff_transcendental_adjoin_image (i : ι) :
AlgebraicIndependent R x ↔ AlgebraicIndependent R (fun j : {j // j ≠ i} ↦ x j) ∧
Transcendental (adjoin R (x '' {i}ᶜ)) (x i) :=
(iff_adjoin_image_compl _).trans <| and_congr_right
fun _ ↦ algebraicIndependent_unique_type_iff (ι := {j // j = i})
variable (hx : AlgebraicIndependent R x)
include hx | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | iff_transcendental_adjoin_image | null |
sumElim {ι'} {y : ι' → A} (hy : AlgebraicIndependent (adjoin R (range x)) y) :
AlgebraicIndependent R (Sum.elim y x) :=
sumElim_iff.mpr ⟨hx, hy⟩ | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | sumElim | null |
sumElim_of_tower {ι'} {y : ι' → A} (hxS : range x ⊆ range (algebraMap S A))
(hy : AlgebraicIndependent S y) : AlgebraicIndependent R (Sum.elim y x) := by
let e := AlgEquiv.ofInjective (IsScalarTower.toAlgHom R S A) hy.algebraMap_injective
set Rx := adjoin R (range x)
let _ : Algebra Rx S :=
(e.symm.toAlgHom.comp <| Subalgebra.inclusion <| adjoin_le hxS).toAlgebra
have : IsScalarTower Rx S A := .of_algebraMap_eq fun x ↦ show _ = (e (e.symm _)).1 by simp; rfl
refine hx.sumElim (hy.restrictScalars (e.symm.injective.comp ?_))
simpa only [AlgHom.coe_toRingHom] using Subalgebra.inclusion_injective _
omit hx in | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | sumElim_of_tower | null |
sumElim_comp {ι'} {x : ι → S} {y : ι' → A} (hx : AlgebraicIndependent R x)
(hy : AlgebraicIndependent S y) : AlgebraicIndependent R (Sum.elim y (algebraMap S A ∘ x)) :=
(hx.map' (f := IsScalarTower.toAlgHom R S A) hy.algebraMap_injective).sumElim_of_tower
(range_comp_subset_range ..) hy | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | sumElim_comp | null |
adjoin_of_disjoint {s t : Set ι} (h : Disjoint s t) :
AlgebraicIndependent (adjoin R (x '' s)) fun i : t ↦ x i :=
((iff_adjoin_image s).mp hx).2.comp (inclusion _) (inclusion_injective h.subset_compl_left) | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | adjoin_of_disjoint | null |
adjoin_iff_disjoint [Nontrivial A] {s t : Set ι} :
(AlgebraicIndependent (adjoin R (x '' s)) fun i : t ↦ x i) ↔ Disjoint s t := by
refine ⟨fun ind ↦ of_not_not fun ndisj ↦ ?_, adjoin_of_disjoint hx⟩
have ⟨i, hs, ht⟩ := Set.not_disjoint_iff.mp ndisj
refine ind.transcendental ⟨i, ht⟩ (isAlgebraic_algebraMap (⟨_, subset_adjoin ?_⟩ : adjoin R _))
exact ⟨i, hs, rfl⟩ | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | adjoin_iff_disjoint | null |
transcendental_adjoin {s : Set ι} {i : ι} (hi : i ∉ s) :
Transcendental (adjoin R (x '' s)) (x i) := by
convert ← hx.adjoin_of_disjoint (Set.disjoint_singleton_right.mpr hi)
rw [algebraicIndependent_singleton_iff ⟨i, rfl⟩] | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | transcendental_adjoin | null |
transcendental_adjoin_iff [Nontrivial A] {s : Set ι} {i : ι} :
Transcendental (adjoin R (x '' s)) (x i) ↔ i ∉ s := by
rw [← Set.disjoint_singleton_right]
convert ← hx.adjoin_iff_disjoint (t := {i})
rw [algebraicIndependent_singleton_iff ⟨i, rfl⟩] | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | transcendental_adjoin_iff | null |
lift_trdeg_add_le [Nontrivial R] [FaithfulSMul R S] [FaithfulSMul S A] :
lift.{v} (trdeg R S) + lift.{u} (trdeg S A) ≤ lift.{u} (trdeg R A) := by
simp_rw [trdeg, lift_iSup (bddAbove_range _)]
simp_rw [Cardinal.ciSup_add_ciSup _ (bddAbove_range _) _ (bddAbove_range _),
add_comm (lift.{v, u} _), ← mk_sum]
refine ciSup_le fun ⟨s, hs⟩ ↦ ciSup_le fun ⟨t, ht⟩ ↦ ?_
have := hs.sumElim_comp ht
refine le_ciSup_of_le (bddAbove_range _) ⟨_, this.to_subtype_range⟩ ?_
rw [← lift_umax, mk_range_eq_of_injective this.injective, lift_id'] | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | lift_trdeg_add_le | null |
trdeg_add_le [Nontrivial R] {A : Type u} [CommRing A] [Algebra R A] [Algebra S A]
[FaithfulSMul R S] [FaithfulSMul S A] [IsScalarTower R S A] :
trdeg R S + trdeg S A ≤ trdeg R A := by
rw [← (trdeg R S).lift_id, ← (trdeg S A).lift_id, ← (trdeg R A).lift_id]
exact lift_trdeg_add_le | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | trdeg_add_le | null |
MvPolynomial.algebraicIndependent_polynomial_aeval_X
(f : ι → Polynomial R) (hf : ∀ i, Transcendental R (f i)) :
AlgebraicIndependent R fun i ↦ Polynomial.aeval (X i : MvPolynomial ι R) (f i) := by
set x := fun i ↦ Polynomial.aeval (X i : MvPolynomial ι R) (f i)
refine algebraicIndependent_of_finite_type' (C_injective _ _) fun t _ _ i hi ↦ ?_
have hle : adjoin R (x '' t) ≤ supported R t := by
rw [Algebra.adjoin_le_iff, Set.image_subset_iff]
intro _ h
rw [Set.mem_preimage]
refine Algebra.adjoin_mono ?_ (Polynomial.aeval_mem_adjoin_singleton R _)
simp_rw [singleton_subset_iff, Set.mem_image_of_mem _ h]
exact (transcendental_supported_polynomial_aeval_X R hi (hf i)).of_tower_top_of_subalgebra_le hle | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | MvPolynomial.algebraicIndependent_polynomial_aeval_X | If for each `i : ι`, `f_i : R[X]` is transcendental over `R`, then `{f_i(X_i) | i : ι}`
in `MvPolynomial ι R` is algebraically independent over `R`. |
AlgebraicIndependent.polynomial_aeval_of_transcendental
(hx : AlgebraicIndependent R x)
{f : ι → Polynomial R} (hf : ∀ i, Transcendental R (f i)) :
AlgebraicIndependent R fun i ↦ Polynomial.aeval (x i) (f i) := by
convert aeval_of_algebraicIndependent hx (algebraicIndependent_polynomial_aeval_X _ hf)
rw [← AlgHom.comp_apply]
congr 1; ext1; simp | theorem | RingTheory | [
"Mathlib.Data.Fin.Tuple.Reflection",
"Mathlib.RingTheory.Algebraic.MvPolynomial",
"Mathlib.RingTheory.AlgebraicIndependent.Basic"
] | Mathlib/RingTheory/AlgebraicIndependent/Transcendental.lean | AlgebraicIndependent.polynomial_aeval_of_transcendental | If `{x_i : A | i : ι}` is algebraically independent over `R`, and for each `i`,
`f_i : R[X]` is transcendental over `R`, then `{f_i(x_i) | i : ι}` is also
algebraically independent over `R`. |
isUnit_of_isIntegral_of_nonZeroDivisor {a : A}
(hi : IsIntegral R a) (ha : a ∈ A⁰) : IsUnit a :=
let B := Algebra.adjoin R {a}
let b : B := ⟨a, Algebra.self_mem_adjoin_singleton R a⟩
haveI : Module.Finite R B := Algebra.finite_adjoin_simple_of_isIntegral hi
haveI : IsArtinianRing B := isArtinian_of_tower R inferInstance
have hinj : Function.Injective B.subtype := Subtype.val_injective
have hb : b ∈ B⁰ := comap_nonZeroDivisors_le_of_injective hinj ha
(isUnit_of_mem_nonZeroDivisors hb).map B.subtype | theorem | RingTheory | [
"Mathlib.RingTheory.Artinian.Module",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/Artinian/Algebra.lean | isUnit_of_isIntegral_of_nonZeroDivisor | In an `R`-algebra over an Artinian ring `R`, if an element is integral and
is not a zero divisor, then it is a unit. |
isUnit_iff_nonZeroDivisor_of_isIntegral {a : A}
(hi : IsIntegral R a) : IsUnit a ↔ a ∈ A⁰ :=
⟨IsUnit.mem_nonZeroDivisors, isUnit_of_isIntegral_of_nonZeroDivisor hi⟩ | theorem | RingTheory | [
"Mathlib.RingTheory.Artinian.Module",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/Artinian/Algebra.lean | isUnit_iff_nonZeroDivisor_of_isIntegral | Integral element of an algebra over Artinian ring `R` is either a zero divisor or a unit. |
isUnit_of_nonZeroDivisor_of_isIntegral' [Algebra.IsIntegral R A] {a : A}
(ha : a ∈ A⁰) : IsUnit a :=
isUnit_of_isIntegral_of_nonZeroDivisor (R := R) (Algebra.IsIntegral.isIntegral a) ha | theorem | RingTheory | [
"Mathlib.RingTheory.Artinian.Module",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/Artinian/Algebra.lean | isUnit_of_nonZeroDivisor_of_isIntegral' | In an `R`-algebra over an Artinian ring `R`, if an element is integral and
is not a zero divisor, then it is a unit. |
isUnit_iff_nonZeroDivisor_of_isIntegral' [Algebra.IsIntegral R A] {a : A} :
IsUnit a ↔ a ∈ A⁰ :=
isUnit_iff_nonZeroDivisor_of_isIntegral (R := R) (Algebra.IsIntegral.isIntegral a) | theorem | RingTheory | [
"Mathlib.RingTheory.Artinian.Module",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/Artinian/Algebra.lean | isUnit_iff_nonZeroDivisor_of_isIntegral' | Integral element of an algebra over Artinian ring `R` is either a zero divisor or a unit. |
isUnit_submonoid_eq_of_isIntegral [Algebra.IsIntegral R A] : IsUnit.submonoid A = A⁰ := by
ext; simpa [IsUnit.mem_submonoid_iff] using isUnit_iff_nonZeroDivisor_of_isIntegral' (R := R) | theorem | RingTheory | [
"Mathlib.RingTheory.Artinian.Module",
"Mathlib.RingTheory.IntegralClosure.Algebra.Defs",
"Mathlib.RingTheory.IntegralClosure.IsIntegral.Basic"
] | Mathlib/RingTheory/Artinian/Algebra.lean | isUnit_submonoid_eq_of_isIntegral | null |
IsArtinian (R M) [Semiring R] [AddCommMonoid M] [Module R M] : Prop :=
WellFoundedLT (Submodule R M) | abbrev | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | IsArtinian | `IsArtinian R M` is the proposition that `M` is an Artinian `R`-module,
implemented as the well-foundedness of submodule inclusion. |
isArtinian_iff (R M) [Semiring R] [AddCommMonoid M] [Module R M] : IsArtinian R M ↔
WellFounded (· < · : Submodule R M → Submodule R M → Prop) :=
isWellFounded_iff _ _ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_iff | null |
LinearMap.isArtinian_iff_of_bijective {S P} [Semiring S] [AddCommMonoid P] [Module S P]
{σ : R →+* S} [RingHomSurjective σ] (l : M →ₛₗ[σ] P) (hl : Function.Bijective l) :
IsArtinian R M ↔ IsArtinian S P :=
let e := Submodule.orderIsoMapComapOfBijective l hl
⟨fun _ ↦ e.symm.strictMono.wellFoundedLT, fun _ ↦ e.strictMono.wellFoundedLT⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | LinearMap.isArtinian_iff_of_bijective | null |
isArtinian_of_injective (f : M →ₗ[R] P) (h : Function.Injective f) [IsArtinian R P] :
IsArtinian R M :=
⟨Subrelation.wf
(fun {A B} hAB => show A.map f < B.map f from Submodule.map_strictMono_of_injective h hAB)
(InvImage.wf (Submodule.map f) IsWellFounded.wf)⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_injective | null |
isArtinian_submodule' [IsArtinian R M] (N : Submodule R M) : IsArtinian R N :=
isArtinian_of_injective N.subtype Subtype.val_injective | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_submodule' | null |
isArtinian_of_le {s t : Submodule R M} [IsArtinian R t] (h : s ≤ t) : IsArtinian R s :=
isArtinian_of_injective (Submodule.inclusion h) (Submodule.inclusion_injective h)
variable (M) in | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_le | null |
isArtinian_of_surjective (f : M →ₗ[R] P) (hf : Function.Surjective f) [IsArtinian R M] :
IsArtinian R P :=
⟨Subrelation.wf
(fun {A B} hAB =>
show A.comap f < B.comap f from Submodule.comap_strictMono_of_surjective hf hAB)
(InvImage.wf (Submodule.comap f) IsWellFounded.wf)⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_surjective | null |
isArtinian_of_surjective_algebraMap {S : Type*} [CommSemiring S] [Algebra S R]
[Module S M] [IsArtinian R M] [IsScalarTower S R M]
(H : Function.Surjective (algebraMap S R)) : IsArtinian S M := by
apply (OrderEmbedding.wellFoundedLT (β := Submodule R M))
refine ⟨⟨?_, ?_⟩, ?_⟩
· intro N
refine {toAddSubmonoid := N.toAddSubmonoid, smul_mem' := ?_}
intro c x hx
obtain ⟨r, rfl⟩ := H c
suffices r • x ∈ N by simpa [Algebra.algebraMap_eq_smul_one, smul_assoc]
apply N.smul_mem _ hx
· intro N1 N2 h
rwa [Submodule.ext_iff] at h ⊢
· intro N1 N2
rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_surjective_algebraMap | If `M` is an Artinian `R` module, and `S` is an `R`-algebra with a surjective
algebra map, then `M` is an Artinian `S` module. |
isArtinian_range (f : M →ₗ[R] P) [IsArtinian R M] : IsArtinian R (LinearMap.range f) :=
isArtinian_of_surjective _ _ f.surjective_rangeRestrict | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_range | null |
isArtinian_of_linearEquiv (f : M ≃ₗ[R] P) [IsArtinian R M] : IsArtinian R P :=
isArtinian_of_surjective _ f.toLinearMap f.toEquiv.surjective | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_linearEquiv | null |
LinearEquiv.isArtinian_iff (f : M ≃ₗ[R] P) : IsArtinian R M ↔ IsArtinian R P :=
⟨fun _ ↦ isArtinian_of_linearEquiv f, fun _ ↦ isArtinian_of_linearEquiv f.symm⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | LinearEquiv.isArtinian_iff | null |
isArtinian_of_finite [Finite M] : IsArtinian R M :=
⟨Finite.wellFounded_of_trans_of_irrefl _⟩
open Submodule | lemma | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_finite | null |
IsArtinian.finite_of_linearIndependent [Nontrivial R] [h : IsArtinian R M] {s : Set M}
(hs : LinearIndependent R ((↑) : s → M)) : s.Finite := by
refine by_contradiction fun hf ↦ (RelEmbedding.wellFounded_iff_isEmpty.1 h.wf).elim' ?_
have f : ℕ ↪ s := Set.Infinite.natEmbedding s hf
have : ∀ n, (↑) ∘ f '' { m | n ≤ m } ⊆ s := by
rintro n x ⟨y, _, rfl⟩
exact (f y).2
have : ∀ a b : ℕ, a ≤ b ↔
span R (Subtype.val ∘ f '' { m | b ≤ m }) ≤ span R (Subtype.val ∘ f '' { m | a ≤ m }) := by
intro a b
rw [span_le_span_iff hs (this b) (this a),
Set.image_subset_image_iff (Subtype.coe_injective.comp f.injective), Set.subset_def]
simp only [Set.mem_setOf_eq]
exact ⟨fun hab x ↦ hab.trans, (· _ le_rfl)⟩
exact ⟨⟨fun n ↦ span R (Subtype.val ∘ f '' { m | n ≤ m }), fun x y ↦ by
rw [le_antisymm_iff, ← this y x, ← this x y]
exact fun ⟨h₁, h₂⟩ ↦ le_antisymm_iff.2 ⟨h₂, h₁⟩⟩, by
intro a b
conv_rhs => rw [GT.gt, lt_iff_le_not_ge, this, this, ← lt_iff_le_not_ge]
rfl⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | IsArtinian.finite_of_linearIndependent | null |
set_has_minimal_iff_artinian :
(∀ a : Set <| Submodule R M, a.Nonempty → ∃ M' ∈ a, ∀ I ∈ a, ¬I < M') ↔ IsArtinian R M := by
rw [isArtinian_iff, WellFounded.wellFounded_iff_has_min] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | set_has_minimal_iff_artinian | A module is Artinian iff every nonempty set of submodules has a minimal submodule among them. |
IsArtinian.set_has_minimal [IsArtinian R M] (a : Set <| Submodule R M) (ha : a.Nonempty) :
∃ M' ∈ a, ∀ I ∈ a, ¬I < M' :=
set_has_minimal_iff_artinian.mpr ‹_› a ha | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | IsArtinian.set_has_minimal | null |
monotone_stabilizes_iff_artinian :
(∀ f : ℕ →o (Submodule R M)ᵒᵈ, ∃ n, ∀ m, n ≤ m → f n = f m) ↔ IsArtinian R M :=
wellFoundedGT_iff_monotone_chain_condition.symm | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | monotone_stabilizes_iff_artinian | A module is Artinian iff every decreasing chain of submodules stabilizes. |
monotone_stabilizes (f : ℕ →o (Submodule R M)ᵒᵈ) : ∃ n, ∀ m, n ≤ m → f n = f m :=
monotone_stabilizes_iff_artinian.mpr ‹_› f | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | monotone_stabilizes | null |
eventuallyConst_of_isArtinian (f : ℕ →o (Submodule R M)ᵒᵈ) :
atTop.EventuallyConst f := by
simp_rw [eventuallyConst_atTop, eq_comm]
exact monotone_stabilizes f | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | eventuallyConst_of_isArtinian | null |
induction {P : Submodule R M → Prop} (hgt : ∀ I, (∀ J < I, P J) → P I) (I : Submodule R M) :
P I :=
WellFoundedLT.induction I hgt
open Function | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | induction | If `∀ I > J, P I` implies `P J`, then `P` holds for all submodules. |
surjective_of_injective_endomorphism (f : M →ₗ[R] M) (s : Injective f) : Surjective f := by
have h := ‹IsArtinian R M›; contrapose! h
rw [IsArtinian, WellFoundedLT, isWellFounded_iff]
refine (RelEmbedding.natGT (LinearMap.range <| f ^ ·) ?_).not_wellFounded
intro n
simp_rw [pow_succ, Module.End.mul_eq_comp, LinearMap.range_comp, ← Submodule.map_top (f ^ n)]
refine Submodule.map_strictMono_of_injective (Module.End.iterate_injective s n) (Ne.lt_top ?_)
rwa [Ne, LinearMap.range_eq_top] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | surjective_of_injective_endomorphism | Any injective endomorphism of an Artinian module is surjective. |
bijective_of_injective_endomorphism (f : M →ₗ[R] M) (s : Injective f) : Bijective f :=
⟨s, surjective_of_injective_endomorphism f s⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | bijective_of_injective_endomorphism | Any injective endomorphism of an Artinian module is bijective. |
disjoint_partial_infs_eventually_top (f : ℕ → Submodule R M)
(h : ∀ n, Disjoint (partialSups (OrderDual.toDual ∘ f) n) (OrderDual.toDual (f (n + 1)))) :
∃ n : ℕ, ∀ m, n ≤ m → f m = ⊤ := by
rsuffices ⟨n, w⟩ : ∃ n : ℕ, ∀ m, n ≤ m → OrderDual.toDual f (m + 1) = ⊤
· use n + 1
rintro (_ | m) p
· cases p
· apply w
exact Nat.succ_le_succ_iff.mp p
obtain ⟨n, w⟩ := monotone_stabilizes (partialSups (OrderDual.toDual ∘ f))
refine ⟨n, fun m p ↦ (h m).eq_bot_of_ge <| sup_eq_left.mp ?_⟩
simpa only [partialSups_add_one] using (w (m + 1) <| le_add_right p).symm.trans <| w m p | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | disjoint_partial_infs_eventually_top | A sequence `f` of submodules of an Artinian module,
with the supremum `f (n+1)` and the infimum of `f 0`, ..., `f n` being ⊤,
is eventually ⊤. |
eventually_iInf_range_pow_eq (f : Module.End R M) :
∀ᶠ n in atTop, ⨅ m, LinearMap.range (f ^ m) = LinearMap.range (f ^ n) := by
obtain ⟨n, hn : ∀ m, n ≤ m → LinearMap.range (f ^ n) = LinearMap.range (f ^ m)⟩ :=
IsArtinian.monotone_stabilizes f.iterateRange
refine eventually_atTop.mpr ⟨n, fun l hl ↦ le_antisymm (iInf_le _ _) (le_iInf fun m ↦ ?_)⟩
rcases le_or_gt l m with h | h
· rw [← hn _ (hl.trans h), hn _ hl]
· exact f.iterateRange.monotone h.le | lemma | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | eventually_iInf_range_pow_eq | null |
isArtinian_of_quotient_of_artinian
(N : Submodule R M) [IsArtinian R M] : IsArtinian R (M ⧸ N) :=
isArtinian_of_surjective M (Submodule.mkQ N) (Submodule.Quotient.mk_surjective N) | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_quotient_of_artinian | null |
isArtinian_of_range_eq_ker [IsArtinian R M] [IsArtinian R P] (f : M →ₗ[R] N) (g : N →ₗ[R] P)
(h : LinearMap.range f = LinearMap.ker g) : IsArtinian R N :=
wellFounded_lt_exact_sequence (LinearMap.range f)
(Submodule.map ((LinearMap.ker f).liftQ f le_rfl))
(Submodule.comap ((LinearMap.ker f).liftQ f le_rfl))
(Submodule.comap g.rangeRestrict) (Submodule.map g.rangeRestrict)
(Submodule.gciMapComap <| LinearMap.ker_eq_bot.mp <| Submodule.ker_liftQ_eq_bot _ _ _ le_rfl)
(Submodule.giMapComap g.surjective_rangeRestrict)
(by simp [Submodule.map_comap_eq, inf_comm, Submodule.range_liftQ])
(by simp [Submodule.comap_map_eq, h]) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_range_eq_ker | null |
isArtinian_iff_submodule_quotient (S : Submodule R P) :
IsArtinian R P ↔ IsArtinian R S ∧ IsArtinian R (P ⧸ S) := by
refine ⟨fun h ↦ ⟨inferInstance, inferInstance⟩, fun ⟨_, _⟩ ↦ ?_⟩
apply isArtinian_of_range_eq_ker S.subtype S.mkQ
rw [Submodule.ker_mkQ, Submodule.range_subtype] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_iff_submodule_quotient | null |
isArtinian_prod [IsArtinian R M] [IsArtinian R P] : IsArtinian R (M × P) :=
isArtinian_of_range_eq_ker (LinearMap.inl R M P) (LinearMap.snd R M P) (LinearMap.range_inl R M P) | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_prod | null |
isArtinian_sup (M₁ M₂ : Submodule R P) [IsArtinian R M₁] [IsArtinian R M₂] :
IsArtinian R ↥(M₁ ⊔ M₂) := by
have := isArtinian_range (M₁.subtype.coprod M₂.subtype)
rwa [LinearMap.range_coprod, Submodule.range_subtype, Submodule.range_subtype] at this
variable {ι : Type*} [Finite ι] | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_sup | null |
isArtinian_pi :
∀ {M : ι → Type*} [Π i, AddCommGroup (M i)]
[Π i, Module R (M i)] [∀ i, IsArtinian R (M i)], IsArtinian R (Π i, M i) := by
apply Finite.induction_empty_option _ _ _ ι
· exact fun e h ↦ isArtinian_of_linearEquiv (LinearEquiv.piCongrLeft R _ e)
· infer_instance
· exact fun ih ↦ isArtinian_of_linearEquiv (LinearEquiv.piOptionEquivProd R).symm | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_pi | null |
isArtinian_pi' [IsArtinian R M] : IsArtinian R (ι → M) :=
isArtinian_pi | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_pi' | A version of `isArtinian_pi` for non-dependent functions. We need this instance because
sometimes Lean fails to apply the dependent version in non-dependent settings (e.g., it fails to
prove that `ι → ℝ` is finite dimensional over `ℝ`). |
isArtinian_finsupp [IsArtinian R M] : IsArtinian R (ι →₀ M) :=
isArtinian_of_linearEquiv (Finsupp.linearEquivFunOnFinite _ _ _).symm | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_finsupp | null |
isArtinian_iSup :
∀ {M : ι → Submodule R P} [∀ i, IsArtinian R (M i)], IsArtinian R ↥(⨆ i, M i) := by
apply Finite.induction_empty_option _ _ _ ι
· intro _ _ e h _ _; rw [← e.iSup_comp]; apply h
· intros; rw [iSup_of_empty]; infer_instance
· intro _ _ ih _ _; rw [iSup_option]; infer_instance
variable (R M) in | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_iSup | null |
IsArtinian.isSemisimpleModule_iff_jacobson [IsArtinian R M] :
IsSemisimpleModule R M ↔ Module.jacobson R M = ⊥ :=
⟨fun _ ↦ IsSemisimpleModule.jacobson_eq_bot R M, fun h ↦
have ⟨s, hs⟩ := Finset.exists_inf_le (Subtype.val (p := fun m : Submodule R M ↦ IsCoatom m))
have _ (m : s) : IsSimpleModule R (M ⧸ m.1.1) := isSimpleModule_iff_isCoatom.mpr m.1.2
let f : M →ₗ[R] ∀ m : s, M ⧸ m.1.1 := LinearMap.pi fun m ↦ m.1.1.mkQ
.of_injective f <| LinearMap.ker_eq_bot.mp <| le_bot_iff.mp fun x hx ↦ by
rw [← h, Module.jacobson, Submodule.mem_sInf]
exact fun m hm ↦ hs ⟨m, hm⟩ <| Submodule.mem_finsetInf.mpr fun i hi ↦
(Submodule.Quotient.mk_eq_zero i.1).mp <| congr_fun hx ⟨i, hi⟩⟩
open Submodule Function | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | IsArtinian.isSemisimpleModule_iff_jacobson | null |
eventually_codisjoint_ker_pow_range_pow (f : Module.End R M) :
∀ᶠ n in atTop, Codisjoint (LinearMap.ker (f ^ n)) (LinearMap.range (f ^ n)) := by
obtain ⟨n, hn : ∀ m, n ≤ m → LinearMap.range (f ^ n) = LinearMap.range (f ^ m)⟩ :=
IsArtinian.monotone_stabilizes f.iterateRange
refine eventually_atTop.mpr ⟨n, fun m hm ↦ codisjoint_iff.mpr ?_⟩
simp_rw [← hn _ hm, Submodule.eq_top_iff', Submodule.mem_sup]
intro x
rsuffices ⟨y, hy⟩ : ∃ y, (f ^ m) ((f ^ n) y) = (f ^ m) x
· exact ⟨x - (f ^ n) y, by simp [hy], (f ^ n) y, by simp⟩
simp_rw [f.pow_apply n, f.pow_apply m, ← iterate_add_apply, ← f.pow_apply (m + n),
← f.pow_apply m, ← mem_range (f := _), ← hn _ (n.le_add_left m), hn _ hm]
exact LinearMap.mem_range_self (f ^ m) x | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | eventually_codisjoint_ker_pow_range_pow | For any endomorphism of an Artinian module, any sufficiently high iterate has codisjoint kernel
and range. |
eventually_isCompl_ker_pow_range_pow [IsNoetherian R M] (f : Module.End R M) :
∀ᶠ n in atTop, IsCompl (LinearMap.ker (f ^ n)) (LinearMap.range (f ^ n)) := by
filter_upwards [f.eventually_disjoint_ker_pow_range_pow.and
f.eventually_codisjoint_ker_pow_range_pow] with n hn
simpa only [isCompl_iff] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | eventually_isCompl_ker_pow_range_pow | This is the Fitting decomposition of the module `M` with respect to the endomorphism `f`.
See also `LinearMap.isCompl_iSup_ker_pow_iInf_range_pow` for an alternative spelling. |
isCompl_iSup_ker_pow_iInf_range_pow [IsNoetherian R M] (f : M →ₗ[R] M) :
IsCompl (⨆ n, LinearMap.ker (f ^ n)) (⨅ n, LinearMap.range (f ^ n)) := by
obtain ⟨k, hk⟩ := eventually_atTop.mp <| f.eventually_isCompl_ker_pow_range_pow.and <|
f.eventually_iInf_range_pow_eq.and f.eventually_iSup_ker_pow_eq
obtain ⟨h₁, h₂, h₃⟩ := hk k (le_refl k)
rwa [h₂, h₃] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isCompl_iSup_ker_pow_iInf_range_pow | This is the Fitting decomposition of the module `M` with respect to the endomorphism `f`.
See also `LinearMap.eventually_isCompl_ker_pow_range_pow` for an alternative spelling. |
range_smul_pow_stabilizes (r : R) :
∃ n : ℕ, ∀ m, n ≤ m →
LinearMap.range (r ^ n • LinearMap.id : M →ₗ[R] M) =
LinearMap.range (r ^ m • LinearMap.id : M →ₗ[R] M) :=
monotone_stabilizes
⟨fun n => LinearMap.range (r ^ n • LinearMap.id : M →ₗ[R] M), fun n m h x ⟨y, hy⟩ =>
⟨r ^ (m - n) • y, by
dsimp at hy ⊢
rw [← smul_assoc, smul_eq_mul, ← pow_add, ← hy, add_tsub_cancel_of_le h]⟩⟩
variable {M} | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | range_smul_pow_stabilizes | null |
exists_pow_succ_smul_dvd (r : R) (x : M) :
∃ (n : ℕ) (y : M), r ^ n.succ • y = r ^ n • x := by
obtain ⟨n, hn⟩ := IsArtinian.range_smul_pow_stabilizes M r
simp_rw [SetLike.ext_iff] at hn
exact ⟨n, by simpa using hn n.succ n.le_succ (r ^ n • x)⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | exists_pow_succ_smul_dvd | null |
isArtinian_of_submodule_of_artinian (R M) [Semiring R] [AddCommMonoid M] [Module R M]
(N : Submodule R M) (_ : IsArtinian R M) : IsArtinian R N := inferInstance | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_submodule_of_artinian | null |
isArtinian_of_tower (R) {S M} [Semiring R] [Semiring S] [AddCommMonoid M] [SMul R S]
[Module S M] [Module R M] [IsScalarTower R S M] (h : IsArtinian R M) : IsArtinian S M :=
⟨(Submodule.restrictScalarsEmbedding R S M).wellFounded h.wf⟩
assert_not_exists IsLocalization IsLocalRing | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_tower | If `M / S / R` is a scalar tower, and `M / R` is Artinian, then `M / S` is also Artinian. |
@[stacks 00J5]
IsArtinianRing (R) [Semiring R] :=
IsArtinian R R | abbrev | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | IsArtinianRing | A ring is Artinian if it is Artinian as a module over itself.
Strictly speaking, this should be called `IsLeftArtinianRing` but we omit the `Left` for
convenience in the commutative case. For a right Artinian ring, use `IsArtinian Rᵐᵒᵖ R`.
For equivalent definitions, see `Mathlib/RingTheory/Artinian/Ring.lean`. |
isArtinianRing_iff {R} [Semiring R] : IsArtinianRing R ↔ IsArtinian R R := Iff.rfl | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinianRing_iff | null |
DivisionSemiring.instIsArtinianRing {K : Type*} [DivisionSemiring K] : IsArtinianRing K :=
⟨Finite.wellFounded_of_trans_of_irrefl _⟩ | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | DivisionSemiring.instIsArtinianRing | null |
DivisionRing.instIsArtinianRing {K : Type*} [DivisionRing K] : IsArtinianRing K :=
inferInstance | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | DivisionRing.instIsArtinianRing | null |
Ring.isArtinian_of_zero_eq_one {R} [Semiring R] (h01 : (0 : R) = 1) : IsArtinianRing R :=
have := subsingleton_of_zero_eq_one h01
inferInstance | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | Ring.isArtinian_of_zero_eq_one | null |
isArtinian_of_fg_of_artinian' {R M} [Ring R] [AddCommGroup M] [Module R M]
[IsArtinianRing R] [Module.Finite R M] : IsArtinian R M :=
have ⟨_, _, h⟩ := Module.Finite.exists_fin' R M
isArtinian_of_surjective _ _ h | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_fg_of_artinian' | null |
isArtinian_of_fg_of_artinian {R M} [Ring R] [AddCommGroup M] [Module R M]
(N : Submodule R M) [IsArtinianRing R] (hN : N.FG) : IsArtinian R N := by
rw [← Module.Finite.iff_fg] at hN; infer_instance | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_of_fg_of_artinian | null |
IsArtinianRing.of_finite (R S) [Ring R] [Ring S] [Module R S] [IsScalarTower R S S]
[IsArtinianRing R] [Module.Finite R S] : IsArtinianRing S :=
isArtinian_of_tower R isArtinian_of_fg_of_artinian' | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | IsArtinianRing.of_finite | null |
isArtinian_span_of_finite (R) {M} [Ring R] [AddCommGroup M] [Module R M] [IsArtinianRing R]
{A : Set M} (hA : A.Finite) : IsArtinian R (Submodule.span R A) :=
isArtinian_of_fg_of_artinian _ (Submodule.fg_def.mpr ⟨A, hA, rfl⟩) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinian_span_of_finite | In a module over an Artinian ring, the submodule generated by finitely many vectors is
Artinian. |
Function.Surjective.isArtinianRing {R} [Semiring R] {S} [Semiring S] {F}
[FunLike F R S] [RingHomClass F R S]
{f : F} (hf : Function.Surjective f) [H : IsArtinianRing R] : IsArtinianRing S := by
rw [isArtinianRing_iff] at H ⊢
exact ⟨(Ideal.orderEmbeddingOfSurjective f hf).wellFounded H.wf⟩ | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | Function.Surjective.isArtinianRing | null |
isArtinianRing_rangeS {R} [Semiring R] {S} [Semiring S] (f : R →+* S) [IsArtinianRing R] :
IsArtinianRing f.rangeS :=
f.rangeSRestrict_surjective.isArtinianRing | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinianRing_rangeS | null |
isArtinianRing_range {R} [Ring R] {S} [Ring S] (f : R →+* S) [IsArtinianRing R] :
IsArtinianRing f.range :=
isArtinianRing_rangeS f | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isArtinianRing_range | null |
RingEquiv.isArtinianRing {R S} [Semiring R] [Semiring S] (f : R ≃+* S)
[IsArtinianRing R] : IsArtinianRing S :=
f.surjective.isArtinianRing | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | RingEquiv.isArtinianRing | null |
isUnit_iff_isRightRegular [IsArtinianRing R] {x : R} : IsUnit x ↔ IsRightRegular x := by
rw [IsRightRegular, IsUnit.isUnit_iff_mulRight_bijective, Bijective, and_iff_left_of_imp]
exact IsArtinian.surjective_of_injective_endomorphism (.toSpanSingleton R R x) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnit_iff_isRightRegular | null |
isUnit_iff_isRegular [IsArtinianRing R] {x : R} : IsUnit x ↔ IsRegular x := by
rw [isRegular_iff, ← isUnit_iff_isRightRegular, and_iff_right_of_imp (·.isRegular.1)] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnit_iff_isRegular | null |
isUnit_iff_isLeftRegular [IsArtinianRing Rᵐᵒᵖ] {x : R} : IsUnit x ↔ IsLeftRegular x := by
rw [← isRightRegular_op, ← isUnit_op, isUnit_iff_isRightRegular] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnit_iff_isLeftRegular | null |
isUnit_iff_isRegular_of_mulOpposite [IsArtinianRing Rᵐᵒᵖ] {x : R} :
IsUnit x ↔ IsRegular x := by
rw [isRegular_iff, ← isUnit_iff_isLeftRegular, and_iff_left_of_imp (·.isRegular.2)] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnit_iff_isRegular_of_mulOpposite | null |
isUnit_of_mem_nonZeroDivisors [IsArtinianRing R] {a : R} (ha : a ∈ R⁰) : IsUnit a := by
rwa [isUnit_iff_isRegular, isRegular_iff_mem_nonZeroDivisors] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnit_of_mem_nonZeroDivisors | If an element of an Artinian ring is not a zero divisor then it is a unit. |
isUnit_of_mem_nonZeroDivisors_of_mulOpposite [IsArtinianRing Rᵐᵒᵖ] {a : R}
(ha : a ∈ R⁰) : IsUnit a := by
rwa [isUnit_iff_isRegular_of_mulOpposite, isRegular_iff_mem_nonZeroDivisors] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnit_of_mem_nonZeroDivisors_of_mulOpposite | null |
isUnit_iff_mem_nonZeroDivisors [IsArtinianRing R] {a : R} : IsUnit a ↔ a ∈ R⁰ := by
rw [isUnit_iff_isRegular, isRegular_iff_mem_nonZeroDivisors] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnit_iff_mem_nonZeroDivisors | In an Artinian ring, an element is a unit iff it is a non-zero-divisor.
See also `isUnit_iff_mem_nonZeroDivisors_of_finite`. |
isUnit_iff_mem_nonZeroDivisors_of_mulOpposite [IsArtinianRing Rᵐᵒᵖ] {a : R} :
IsUnit a ↔ a ∈ R⁰ := by
rw [isUnit_iff_isRegular_of_mulOpposite, isRegular_iff_mem_nonZeroDivisors]
variable (R) | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnit_iff_mem_nonZeroDivisors_of_mulOpposite | null |
isUnitSubmonoid_eq [IsArtinianRing R] : IsUnit.submonoid R = R⁰ := by
ext; simp [IsUnit.mem_submonoid_iff, isUnit_iff_mem_nonZeroDivisors]
@[deprecated (since := "2025-08-26")] alias isUnit_submonoid_eq := isUnitSubmonoid_eq | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnitSubmonoid_eq | null |
isUnitSubmonoid_eq_of_mulOpposite [IsArtinianRing Rᵐᵒᵖ] :
IsUnit.submonoid R = R⁰ := by
ext; simp [IsUnit.mem_submonoid_iff, isUnit_iff_mem_nonZeroDivisors_of_mulOpposite] | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnitSubmonoid_eq_of_mulOpposite | null |
isUnitSubmonoid_eq_nonZeroDivisorsRight [IsArtinianRing R] :
IsUnit.submonoid R = nonZeroDivisorsRight R := by
ext; rw [← isRightRegular_iff_mem_nonZeroDivisorsRight]; exact isUnit_iff_isRightRegular | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isUnitSubmonoid_eq_nonZeroDivisorsRight | null |
nonZeroDivisorsLeft_eq_isUnitSubmonoid [IsArtinianRing Rᵐᵒᵖ] :
IsUnit.submonoid R = nonZeroDivisorsLeft R := by
ext; rw [← isLeftRegular_iff_mem_nonZeroDivisorsLeft]; exact isUnit_iff_isLeftRegular | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | nonZeroDivisorsLeft_eq_isUnitSubmonoid | null |
@[stacks 00J7]
setOf_isMaximal_finite : {I : Ideal R | I.IsMaximal}.Finite := by
have ⟨s, H⟩ := Finset.exists_inf_le (Subtype.val (p := fun I : Ideal R ↦ I.IsMaximal))
refine Set.finite_def.2 ⟨s, fun p ↦ ?_⟩
have ⟨q, hq1, hq2⟩ := p.2.isPrime.inf_le'.mp (H p)
rwa [← Subtype.ext <| q.2.eq_of_le p.2.ne_top hq2] | lemma | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | setOf_isMaximal_finite | null |
isField_of_isDomain [IsDomain R] : IsField R := by
refine ⟨Nontrivial.exists_pair_ne, mul_comm, fun {x} hx ↦ ?_⟩
obtain ⟨n, y, hy⟩ := IsArtinian.exists_pow_succ_smul_dvd x (1 : R)
replace hy : x ^ n * (x * y - 1) = 0 := by
rw [mul_sub, sub_eq_zero]
convert hy using 1
simp [Nat.succ_eq_add_one, pow_add, mul_assoc]
rw [mul_eq_zero, sub_eq_zero] at hy
exact ⟨_, hy.resolve_left <| pow_ne_zero _ hx⟩
/- Does not hold in a commutative semiring:
consider {0, 0.5, 1} with ⊔ as + and ⊓ as *, then both {0} and {0, 0.5} are prime ideals. -/ | lemma | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isField_of_isDomain | null |
isMaximal_of_isPrime {R : Type*} [CommRing R] (p : Ideal R) [p.IsPrime]
[IsArtinianRing R] : p.IsMaximal :=
Ideal.Quotient.maximal_of_isField _ (isField_of_isDomain _) | instance | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isMaximal_of_isPrime | null |
isPrime_iff_isMaximal (p : Ideal R) : p.IsPrime ↔ p.IsMaximal :=
⟨fun _ ↦ isMaximal_of_isPrime p, fun h ↦ h.isPrime⟩ | lemma | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isPrime_iff_isMaximal | null |
@[simps]
primeSpectrumEquivMaximalSpectrum : PrimeSpectrum R ≃ MaximalSpectrum R where
toFun I := ⟨I.asIdeal, isPrime_iff_isMaximal I.asIdeal |>.mp I.isPrime⟩
invFun I := ⟨I.asIdeal, isPrime_iff_isMaximal I.asIdeal |>.mpr I.isMaximal⟩ | def | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | primeSpectrumEquivMaximalSpectrum | The prime spectrum is in bijection with the maximal spectrum. |
primeSpectrumEquivMaximalSpectrum_comp_asIdeal :
MaximalSpectrum.asIdeal ∘ primeSpectrumEquivMaximalSpectrum =
PrimeSpectrum.asIdeal (R := R) := rfl | lemma | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | primeSpectrumEquivMaximalSpectrum_comp_asIdeal | null |
primeSpectrumEquivMaximalSpectrum_symm_comp_asIdeal :
PrimeSpectrum.asIdeal ∘ primeSpectrumEquivMaximalSpectrum.symm =
MaximalSpectrum.asIdeal (R := R) := rfl | lemma | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | primeSpectrumEquivMaximalSpectrum_symm_comp_asIdeal | null |
primeSpectrum_asIdeal_range_eq :
range PrimeSpectrum.asIdeal = (range <| MaximalSpectrum.asIdeal (R := R)) := by
simp only [PrimeSpectrum.range_asIdeal, MaximalSpectrum.range_asIdeal,
isPrime_iff_isMaximal]
variable (R) | lemma | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | primeSpectrum_asIdeal_range_eq | null |
setOf_isPrime_finite : {I : Ideal R | I.IsPrime}.Finite := by
simpa only [isPrime_iff_isMaximal] using setOf_isMaximal_finite R | lemma | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | setOf_isPrime_finite | null |
@[local instance] noncomputable fieldOfSubtypeIsMaximal
(I : MaximalSpectrum R) : Field (R ⧸ I.asIdeal) :=
Ideal.Quotient.field I.asIdeal | def | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | fieldOfSubtypeIsMaximal | A temporary field instance on the quotients by maximal ideals. |
noncomputable quotNilradicalEquivPi :
R ⧸ nilradical R ≃+* ∀ I : MaximalSpectrum R, R ⧸ I.asIdeal :=
let f := MaximalSpectrum.asIdeal (R := R)
.trans
(Ideal.quotEquivOfEq <| ext fun x ↦ by
rw [PrimeSpectrum.nilradical_eq_iInf, iInf, primeSpectrum_asIdeal_range_eq]; rfl)
(Ideal.quotientInfRingEquivPiQuotient f <| fun I J h ↦
Ideal.isCoprime_iff_sup_eq.mpr <| I.2.coprime_of_ne J.2 <|
fun hIJ ↦ h <| MaximalSpectrum.ext hIJ) | def | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | quotNilradicalEquivPi | The quotient of a commutative Artinian ring by its nilradical is isomorphic to
a finite product of fields, namely the quotients by the maximal ideals. |
noncomputable equivPi [IsReduced R] : R ≃+* ∀ I : MaximalSpectrum R, R ⧸ I.asIdeal :=
.trans (.symm <| .quotientBot R) <| .trans
(Ideal.quotEquivOfEq (nilradical_eq_zero R).symm) (quotNilradicalEquivPi R) | def | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | equivPi | A reduced commutative Artinian ring is isomorphic to a finite product of fields,
namely the quotients by the maximal ideals. |
isSemisimpleRing_of_isReduced [IsReduced R] : IsSemisimpleRing R :=
(equivPi R).symm.isSemisimpleRing | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isSemisimpleRing_of_isReduced | null |
isSemisimpleRing_iff_jacobson : IsSemisimpleRing R ↔ Ring.jacobson R = ⊥ :=
IsArtinian.isSemisimpleModule_iff_jacobson R R | theorem | RingTheory | [
"Mathlib.Algebra.Group.Units.Opposite",
"Mathlib.Algebra.Regular.Opposite",
"Mathlib.Data.SetLike.Fintype",
"Mathlib.Order.Filter.EventuallyConst",
"Mathlib.RingTheory.Ideal.Prod",
"Mathlib.RingTheory.Ideal.Quotient.Operations",
"Mathlib.RingTheory.Jacobson.Semiprimary",
"Mathlib.RingTheory.Nilpotent.... | Mathlib/RingTheory/Artinian/Module.lean | isSemisimpleRing_iff_jacobson | null |
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